We present observations of escaping Lyman Continuum (LyC) radiation from 34 massive star-forming galaxies (SFGs) and 12 weak AGN with reliably measured spectroscopic redshifts at $z\simeq 2.3\mbox{--}4.1$. We analyzed Hubble Space Telescope (HST) Wide Field Camera 3 (WFC3) mosaics of the Early Release Science (ERS) field in three UVIS filters to sample the rest-frame LyC over this redshift range. With our best current assessment of the WFC3 systematics, we provide $1\sigma $ upper limits for the average LyC emission of galaxies at $\langle z\rangle $ = 2.35, 2.75, and 3.60 to ~28.5, 28.1, and 30.7 mag in image stacks of 11–15 galaxies in the WFC3/UVIS F225W, F275W, and F336W, respectively. The LyC flux of weak AGN at $\langle z\rangle $ = 2.62 and 3.32 are detected at 28.3 and 27.4 mag with S/Ns of ~2.7 and 2.5 in F275W and F336W for stacks of 7 and 3 AGN, respectively, while AGN at $\langle z\rangle $ = 2.37 are constrained to gsim27.9 mag at $1\sigma $ in a stack of 2 AGN. The stacked AGN LyC light profiles are flatter than their corresponding non-ionizing UV continuum profiles out to radii of $r\lesssim 0\buildrel{\prime\prime}\over{.} 9$, which may indicate a radial dependence of porosity in the ISM. With synthetic stellar SEDs fit to UV continuum measurements longward of ${\rm{Ly}}\alpha $ and IGM transmission models, we constrain the absolute LyC escape fractions to ${f}_{\mathrm{esc}}^{\mathrm{abs}}\simeq {22}_{-22}^{+44}$% at $\langle z\rangle $ = 2.35 and lesssim55% at $\langle z\rangle $ = 2.75 and 3.60, respectively. All available data for galaxies, including published work, suggests a more sudden increase of ${f}_{\mathrm{esc}}$ with redshift at $z\simeq 2$. Dust accumulating in (massive) galaxies over cosmic time correlates with increased H i column density, which may lead to reducing ${f}_{\mathrm{esc}}$ more suddenly at $z\lesssim 2$. This may suggest that SFGs collectively contributed to maintaining cosmic reionization at redshifts $z\gtrsim 2\mbox{--}4$, while AGN likely dominated reionization at $z\lesssim 2$.